Question

Prove that the following identity is true.

sin^4t − cos^4t /sin^2t cos^2t = sec^2 t − csc^2 t

Answer 1

Let us begin from LHS
LHS= sin^4t − cos^4t /sin^2t cos^2t
=(sin^2t − cos^2t)(sin^2t + cos^2t) /sin^2t cos^2t
=(sin^2t − cos^2t)*1 /sin^2t cos^2t=(sin^2t − cos^2t) /sin^2t cos^2t
=sin^2t/sin^2t cos^2t − cos^2t/sin^2t cos^2t
=1/cos^2t − 1/sin^2t
= sec^2 t − csc^2 t
=RHS
hence the given identity is true.

Answer 2

@dpasingh I'm confused on your second line. is it \[\sin^2t-\cos^2t ( \frac{ 1 }{ \sin^2tcos^2t })\] ?

Answer 3

@itsonlycdeee
Here i have divided each term of i.e. \[\frac{(\sin^2t − \cos^2t)}{ \sin^2t \cos^2t}=\frac{\sin^2t}{ \sin^2t \cos^2t}-\frac{\cos^2t}{ \sin^2t \cos^2t}\]
\[=\frac{1}{ \cos^2t}-\frac{1}{\sin^2t }=\sec^2 t - cosec^2 t\]

Answer 4

@dpasingh okay thank you! that it made it more clearer.